Questions: Concept Simulation 10.3 illustrates the concepts pertinent to this problem. A 0.47-kg object is attached to one end of a spring, as in the first drawing, and the system is set into simple harmonic motion. The displacement x of the object as a function of time is shown in the second drawing. With the aid of there data, determine (a) the amplitude A of the motion, (b) the angular frequency ω, (c) the spring constant k, (d) the speed of the object at t=1.0 s, and (e) the magnitude of the object's acceleration at t=1.0 s.

Solution Steps

The amplitude \( A \) is the maximum displacement from the equilibrium position in simple harmonic motion. From the displacement-time graph, identify the maximum value of \( x \). This value is the amplitude \( A \).

The angular frequency \( \omega \) is related to the period \( T \) of the motion by the formula: \[ \omega = \frac{2\pi}{T} \] Determine the period \( T \) from the graph by measuring the time it takes for the object to complete one full cycle of motion.

The spring constant \( k \) can be found using the formula for angular frequency in terms of mass \( m \) and spring constant \( k \): \[ \omega = \sqrt{\frac{k}{m}} \] Rearrange to solve for \( k \): \[ k = m\omega^2 \] Substitute the values of \( m \) and \( \omega \) to find \( k \).

Final Answer